Answer:
a) 
  
Where  and
 and  
b)  
And we can find this probability using the complement rule and with the normal standard table or excel:
 
c)  
 
d)  
 
 
e)  IQR = 75.13-50.87=24.26
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  
Part a
Let X the random variable that represent the amount of time that people spend at Grover Hot Springs of a population, and for this case we know the distribution for X is given by:
 
  
Where  and
 and  
Part b
We are interested on this probability
 
And the best way to solve this problem is using the normal standard distribution and the z score given by:
 
If we apply this formula to our probability we got this:
 
And we can find this probability using the complement rule and with the normal standard table or excel:
 
Part c
We are interested on this probability
 
And the best way to solve this problem is using the normal standard distribution and the z score given by:
 
If we apply this formula to our probability we got this:
 
And we can find this probability using the normal standard table or excel:
 
Part d
If we apply this formula to our probability we got this:
 
And we can find this probability with this difference:
 
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  
 
Part e
Q1
For this part we want to find a value a, such that we satisfy this condition:
 (a)
   (a)
 (b)
   (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  
As we can see on the figure attached the z value that satisfy the condition with 0.25 of the area on the left and 0.75 of the area on the right it's z=-0.674. On this case P(Z<-0.674)=0.25 and P(z>-0.674)=0.75
If we use condition (b) from previous we have this:
 
  
 
But we know which value of z satisfy the previous equation so then we can do this:
 
And if we solve for a we got
 
So the value of height that separates the bottom 25% of data from the top 75% is 50.87.  
Q3
Since the distribution is symmetrical we repaeat the procedure for Q1 but now with z= 0.674
 
And if we solve for a we got
 
And then the IQR = 75.13-50.87=24.26